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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Self-adjoint Jacobi matrices on trees and multiple orthogonal polynomials
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by Alexander I. Aptekarev, Sergey A. Denisov and Maxim L. Yattselev PDF
Trans. Amer. Math. Soc. 373 (2020), 875-917 Request permission

Abstract:

We consider a set of measures on the real line and the corresponding system of multiple orthogonal polynomials (MOPs) of the first and second type. Under some very mild assumptions, which are satisfied by Angelesco systems, we define self-adjoint Jacobi matrices on certain rooted trees. We express their Green’s functions and the matrix elements in terms of MOPs. This provides a generalization of the well-known connection between the theory of polynomials orthogonal on the real line and Jacobi matrices on $\mathbb {Z}_+$ to a higher dimension. We illustrate the importance of this connection by proving ratio asymptotics for MOPs using methods of operator theory.
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Additional Information
  • Alexander I. Aptekarev
  • Affiliation: Keldysh Institute of Applied Mathematics, Russian Academy of Science, Moscow, Russian Federation
  • MR Author ID: 192572
  • Email: aptekaa@keldysh.ru
  • Sergey A. Denisov
  • Affiliation: Department of Mathematics, University of Wisconsin–Madison, 480 Lincoln Drive, Madison, Wisconsin 53706
  • MR Author ID: 627554
  • Email: denissov@math.wisc.edu
  • Maxim L. Yattselev
  • Affiliation: Department of Mathematical Sciences, Indiana University-Purdue University Indianapolis, 402 North Blackford Street, Indianapolis, Indiana 46202
  • MR Author ID: 789878
  • Email: maxyatts@iupui.edu
  • Received by editor(s): June 27, 2018
  • Received by editor(s) in revised form: January 15, 2019
  • Published electronically: October 28, 2019
  • Additional Notes: The research of the first author was carried out with support from a grant of the Russian Science Foundation (project RScF-14-21-00025)
    The work of the second author done in the last section of the paper was supported by a grant of the Russian Science Foundation (project RScF-14-21-00025), and his research on the rest of the paper was supported by the grant NSF-DMS-1464479 and a Van Vleck Professorship Research Award
    The research of the third author was supported in part by a grant from the Simons Foundation, CGM-354538
  • © Copyright 2019 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 373 (2020), 875-917
  • MSC (2010): Primary 42C05, 47B36
  • DOI: https://doi.org/10.1090/tran/7959
  • MathSciNet review: 4068253